# How to Break Down a Composition of Functions

A *composition* of functions is one function acting upon another. Think of it like putting one function inside of the other — *f*(*g*(*x*)), for instance, means that you plug the entire *g*(*x*) function in for all *x*’s in *f*(*x*). To solve such a problem, you work from the inside out:

*f*(*g*(*x*)) = *f*(3*x*^{2} – 10) = (3*x*^{2} – 10)^{2} – 6(3*x*^{2} – 10) + 1

This process puts the *g*(*x*) function into the *f*(*x*) function everywhere the *f*(*x*) function asks for *x.* This equation ultimately simplifies to 9*x*^{4} – 78*x*^{2} + 161, in case you’re asked to simplify the composition (which you usually are).

Likewise,

which easily simplifies to 3(2*x* – 1) – 10 because the square root and square cancel each other. This equation simplifies even further to 6*x* – 13.

You may also be asked to find one value of a composed function. To find

for instance, it helps to realize that it’s like reading Hebrew: You work from right to left. In this example, you’re asked to put –3 in for *x* in *f*(*x*), get an answer, and then plug that answer in for *x* in *g*(*x*). Here are these two steps in action:

*f*(–3) = (–3)^{2} – 6(–3) + 1 = 28

*g*(28) = 3(28)^{2} – 10 = 2,342